Analytic and geometric properties of open door functions

In this paper, we study analytic and geometric properties of the solution $q(z)$ to the differential equation $q(z)+zq'(z)/q(z)=h(z)$ with the initial condition $q(0)=1$ for a given analytic function $h(z)$ on the unit disk $|z|<1$ in the complex plane with $h(0)=1.$ In particular, we investigate the possible largest constant $c>0$ such that the condition $|\Im[zf"(z)/f'(z)]|<c$ on $|z|<1$ implies starlikeness of an analytic function $f(z)$ on $|z|<1$ with $f(0)=f'(0)-1=0.$